Dating the Break in High-dimensional Data

TL;DR

This paper introduces a new high-dimensional change point estimator based on U-statistics, offering improved efficiency, asymptotic properties, and validated confidence intervals through simulations and bootstrap methods.

Contribution

It develops a novel U-statistic based estimator for change point detection in high-dimensional data with superior efficiency and new asymptotic theory, including bootstrap validity.

Findings

The new estimator outperforms least squares methods in simulations.

Bootstrap confidence intervals are valid and competitive.

Asymptotic theory for high-dimensional U-statistics is established.

Abstract

This paper is concerned with estimation and inference for the location of a change point in the mean of independent high-dimensional data. Our change point location estimator maximizes a new U-statistic based objective function, and its convergence rate and asymptotic distribution after suitable centering and normalization are obtained under mild assumptions. Our estimator turns out to have better efficiency as compared to the least squares based counterpart in the literature. Based on the asymptotic theory, we construct a confidence interval by plugging in consistent estimates of several quantities in the normalization. We also provide a bootstrap-based confidence interval and state its asymptotic validity under suitable conditions. Through simulation studies, we demonstrate favorable finite sample performance of the new change point location estimator as compared to its least squares based counterpart, and our bootstrap-based confidence intervals, as compared to several existing competitors. The asymptotic theory based on high-dimensional U-statistic is substantially different from those developed in the literature and is of independent interest.